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Boundary value problems for the Dunkl Laplacian

Chrouda M. B., Mabrouk K. E., Hassine K.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 2

249--269

Let Δk be the Dunkl Laplacian on Rd associated with a reflection group W and a multiplicity function k. The purpose of this paper is to establish the existence and the uniqueness of a positive solution on the unit ball B of Rd to the following boundary value problem: Δku = φ(u) in B and u = f on ∂B: We distinguish two cases of nonnegative perturbation φ: trivial and nontrivial.

One-dimensional symmetric stable Feynman-Kac semigroups

Byczkowska H., Byczkowski T.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

381--404

We investigate here one-dimensional Feynman-Kac semigroups based on symmetric α-stable processes. We begin with establishing the properties of Green operators of intervals and halflines on functions from the Kato class. Then we provide a sufficient condition for gaugeability of the halfline(−∞, b) and evaluate the critical value β.